Spacetime Topology Change |

**In General**

* __Motivation__: (1) In
a path-integral formulation of quantum gravity, we would like to sum over all
metrics, but also over all different topologies, interpolating between two given
manifolds; (2) Possibility of creating monopole-antimonopole pairs; (3) Validity
of the spin-stats theorem; (4) Possibility of getting fermions and internal
symm multiplets in pure general relativity; (5) Allow second quantization of
geons (consistency); (6) Quantum topology change at small scales would cost
little action.

* __Early ideas__: Jordan
(different spaces may unite–astronomically motivated).

* __Criteria__: We want to
exclude models with infinite particle production, and possibly also those with
double light cones; It might be possible to ensure this by requiring continuity
of time as volume of past light cone.

* __Mechanism__: It would
presumably be a quantum phenomenon, occurring only at microscopic scales, since
there are no classical topology-changing solutions in general relativity; In
quantum gravity one could get changing amplitudes for various topologies; One
possibility however is through some modification of the Einstein equation.

* __Controversy__: DeWitt & Anderson,
Castagnino, Dray & Manogue (not ok, infinite particle production in trousers);
Sorkin et al (ok, but in higher dimensions some elementary cobordisms might not
be equally suppressed).

**Kinematics: Cobordism**
> s.a. models of topology change.

* __Idea__: We require the existence
of an interpolating manifold between two given spatial geometries (topological
cobordism), on which we can then put a metric (Riemannian, Lorentzian or causal
cobordism).

* __Topological__: It always exists
for pair creation (e.g., in 3-dimensions, creating two geons of the kind
\(\mathbb R\)P^{2} # \(\mathbb R^2\)
–non-orientable– or T^{2}
# \(\mathbb R\)^{2}–orientable);
More generally it exists if the initial and final manifolds are cobordant,
which happens iff their Stiefel-Whitney numbers are equal; One can use surgery
to obtain the desired cobordism (allows Δ*χ* = ±2 for
*n* > 3), a cobordism is like a sequence of localized surgeries.

* __Riemannian__: Given a topological
one, a Riemannian cobordism is always possible.

* __Lorentzian__: If
the manifold is time-orientable, it is possible only if we allow
the metric to have closed timelike curves (likely to be very small,
for dynamical reasons); __Conditions__: if ∂*M*
= *M*_{1} ∪
*M*_{2}, in even
dimensions, *χ*(*M*) = 0; In odd dimensions,
*χ*(*M*_{1})
= *χ*(*M*_{2});
There is no possible Lorentzian topology change in 0+1, 1+1 and 2+1 dimensions.

* __Causal__: We require
no causality violations, but allow the metric to be singular (= 0)
at isolated points:

- Pair creation: In even
(> 1+1) dimensions it can always be obtained; In 4+1 Kaluza-Klein
monopole-antimonopole pairs can be created (with non time-orientable metrics).

- Local causality structure: In 1+1
dimensions both future and past light cones of singular points are double;
In 2+1 only one of them need split; In 3+1 neither.

@ __General references__:
Treder AdP(62);
Kreisel et al AdP(63) [degenerate];
Crampin PCPS(68);
Antonelli & Williams IJTP(79) [and kink field theories];
Borde gq/94.

@ __Degenerate metrics, causality__:
Horowitz CQG(91);
Louko & Sorkin CQG(97)gq/95 [complex action];
Matschull CQG(96)gq/95;
Borde et al CQG(99)gq.

**Phenomenology** > s.a. models of topology change;
wormholes [scale-dependent topology].

@ __General references__: Tanaka & Nagami IJGMP(13) [dark-matter production];
Antoniou et al a1812 [and surgery, wormholes].

@ __And quantum coherence__: Coleman NPB(88);
Lavrelashvili et al NPB(88).

@ __And black-hole information, unitarity__:
Barbón & Rabinovici ht/05-conf;
Hsu PLB(07)ht/06 [baby universes].

**References**
> s.a. Cobordism; models
of topology change; spacetime foam.

@ __Intros, reviews__: in Sorkin in(90);
Gibbons in(92)-a1110,
in(93);
Callender & Weingard SHPMP(00) [conceptual];
Dowker gq/02-proc;
Asorey et al a1211.

@ __General__: Misner & Wheeler AP(57);
Geroch JMP(67);
Brill in(72);
Yodzis CMP(72),
GRG(73);
Tipler PRL(76),
AP(77);
Lee PRS(78);
Strominger PRL(84);
Konstantinov & Melnikov CQG(86);
Sorkin PRD(86) [conditions, and monople creation];
Anderson PLB(88);
Banks NPB(88);
De Ritis et al NCB(88);
Visser PRD(90);
Horowitz CQG(91);
Gibbons & Hawking CMP(92),
PRL(92);
Borde gq/94;
del Campo PRD(95);
Konstantinov IJMPD(98)gq/95;
Borowiec et al IJGMP(07) [Lagrangian formalism].

@ __And causal continuity__: Dowker & Surya PRD(98)gq/97;
Dowker et al CQG(00)gq/99.

@ __And cosmic censorship__:
Joshi & Saraykar PLA(87);
Etesi a1905 [strong cosmic censorship violations].

@ __Related topics__: Komorowski pr(71) [topology on superspace];
Gibbons CQG(93) [and matter fields, skyrmions];
Maia IJMPCS(12)-a1211 [and the cosmological constant].

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